Covariance

In probability theory and statistics, covariance is a measure of the joint variability of two random variables.[1] If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values, (i.e., the variables tend to show similar behavior), the covariance is positive.[2] In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (i.e., the variables tend to show opposite behavior), the covariance is negative. The sign of the covariance therefore shows the tendency in the linear relationship between the variables. The magnitude of the covariance is not easy to interpret because it is not normalized and hence depends on the magnitudes of the variables. The normalized version of the covariance, the correlation coefficient, however, shows by its magnitude the strength of the linear relation.

A distinction must be made between (1) the covariance of two random variables, which is a populationparameter that can be seen as a property of the joint probability distribution, and (2) the sample covariance, which in addition to serving as a descriptor of the sample, also serves as an estimated value of the population parameter.

where E[X] is the expected value of X, also known as the mean of X. The covariance is also sometimes denoted σXY or σ(X,Y), in analogy to variance. By using the linearity property of expectations, this can be simplified to the expected value of their product minus the product of their expected values:

However, when E⁡[XY]≈E⁡[X]E⁡[Y]{\displaystyle \operatorname {E} [XY]\approx \operatorname {E} [X]\operatorname {E} [Y]}, this last equation is prone to catastrophic cancellation when computed with floating point arithmetic and thus should be avoided in computer programs when the data has not been centered before.[4]Numerically stable algorithms should be preferred in this case.

The (i, j)-th element of this matrix is equal to the covariance cov(Xi, Yj) between the i-th scalar component of X and the j-th scalar component of Y. In particular, cov(Y, X) is the transpose of cov(X, Y).

For a vector X=[X1X2…Xm]T{\displaystyle \mathbf {X} ={\begin{bmatrix}X_{1}&X_{2}&\dots &X_{m}\end{bmatrix}}^{\mathrm {T} }} of m jointly distributed random variables with finite second moments, its covariance matrix (also known as the variance–covariance matrix) is defined as

Random variables whose covariance is zero are called uncorrelated. Similarly, the components of random vectors whose covariance matrix is zero in every entry outside the main diagonal are also called uncorrelated.

The units of measurement of the covariance cov(X, Y) are those of X times those of Y. By contrast, correlation coefficients, which depend on the covariance, are a dimensionless measure of linear dependence. (In fact, correlation coefficients can simply be understood as a normalized version of covariance.)

If the random variable pair (X, Y) can take on the values (xi, yi) for i = 1, ... , n, with equal probabilities 1/n, then the covariance can be equivalently written in terms of the means E(X){\displaystyle E(X)} and E(Y){\displaystyle E(Y)} as

Suppose that X and Y have the following joint probability mass function,[6] in which the six central cells give the probabilities f(x, y) of the six hypothetical realizations (x, y) = (1, 1), (1, 2), (1, 3), (2, 1), (2,2), and (2, 3):

y

f(x,y)

1

2

3

fX(x)

1

1/4

1/4

0

1/2

x

2

0

1/4

1/4

1/2

fY(y)

1/4

1/2

1/4

1

X can take on two values (1 and 2) while Y can take on three (1, 2, and 3). Their means are μX=3/2{\displaystyle \mu _{X}=3/2} and μY=2.{\displaystyle \mu _{Y}=2.} The population standard deviations of X and Y are σX=1/2{\displaystyle \sigma _{X}=1/2} and σY=1/2.{\displaystyle \sigma _{Y}={\sqrt {1/2}}.} Then:

If X, Y, W, and V are real-valued random variables and a, b, c, d are constant ("constant" in this context means non-random), then the following facts are a consequence of the definition of covariance:

In this case, the relationship between Y and X is non-linear, while correlation and covariance are measures of linear dependence between two variables. This example shows that if two variables are uncorrelated, that does not in general imply that they are independent. However, if two variables are jointly normally distributed (but not if they are merely individually normally distributed), uncorrelatedness does imply independence.

In fact these properties imply that the covariance defines an inner product over the quotient vector space obtained by taking the subspace of random variables with finite second moment and identifying any two that differ by a constant. (This identification turns the positive semi-definiteness above into positive definiteness.) That quotient vector space is isomorphic to the subspace of random variables with finite second moment and mean zero; on that subspace, the covariance is exactly the L2 inner product of real-valued functions on the sample space.

As a result, for random variables with finite variance, the inequality

The sample covariances among K variables based on N observations of each, drawn from an otherwise unobserved population, are given by the K-by-Kmatrixq¯=[qjk]{\displaystyle \textstyle {\overline {\mathbf {q} }}=\left[q_{jk}\right]} with the entries

which is an estimate of the covariance between variable j and variable k.

The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vectorX{\displaystyle \textstyle \mathbf {X} }, a vector whose jth element (j = 1, ..., K) is one of the random variables. The reason the sample covariance matrix has N−1{\displaystyle \textstyle N-1} in the denominator rather than N{\displaystyle \textstyle N} is essentially that the population mean E⁡(X){\displaystyle \operatorname {E} (X)} is not known and is replaced by the sample mean X¯{\displaystyle \mathbf {\bar {X}} }. If the population mean E⁡(X){\displaystyle \operatorname {E} (X)} is known, the analogous unbiased estimate is given by

The covariance is sometimes called a measure of "linear dependence" between the two random variables. That does not mean the same thing as in the context of linear algebra (see linear dependence). When the covariance is normalized, one obtains the Pearson correlation coefficient, which gives the goodness of the fit for the best possible linear function describing the relation between the variables. In this sense covariance is a linear gauge of dependence.

Covariance is an important measure in biology. Certain sequences of DNA are conserved more than others among species, and thus to study secondary and tertiary structures of proteins, or of RNA structures, sequences are compared in closely related species. If sequence changes are found or no changes at all are found in noncoding RNA (such as microRNA), sequences are found to be necessary for common structural motifs, such as an RNA loop. In genetics, covariance serves a basis for computation of Genetic Relationship Matrix (GRM) (aka kinship matrix), enabling inference on population structure from sample with no known close relatives as well as inference on estimation of heritability of complex traits.

The covariance matrix is important in estimating the initial conditions required for running weather forecast models. The 'forecast error covariance matrix' is typically constructed between perturbations around a mean state (either a climatological or ensemble mean). The 'observation error covariance matrix' is constructed to represent the magnitude of combined observational errors (on the diagonal) and the correlated errors between measurements (off the diagonal). This is an example of its widespread application to Kalman filtering and more general state estimation for time-varying systems.

The eddy covariance technique is a key atmospherics measurement technique where the covariance between instantaneous deviation in vertical wind speed from the mean value and instantaneous deviation in gas concentration is the basis for calculating the vertical turbulent fluxes.

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